Lattice (order)

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In mathematics, a lattice is a partially ordered set (or poset) whose nonempty finite subsets all have a supremum (called join) and an infimum (called meet). Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions.

Lattices as posets

Consider a poset (L, ≤). L is a lattice if

For all elements x and y of L, the set has both a least upper bound in L (join, or supremum) and a greatest lower bound in L (meet, or infimum).

The join and meet of x and y are denoted by [x \vee y] and [x \wedge y], respectively. Because joins and meets are assumed to exist in a lattice, [\vee] and [\wedge] are binary operations. Hence this definition is equivalent to requiring L to be both a join- and a meet-semilattice.

A bounded lattice has a greatest and least element, denoted 1 and 0 by convention (also called top and bottom). Any lattice can be converted into a bounded lattice by adding a greatest and least element.

Using an easy induction argument, one can deduce the existence of suprema (joins) and infima (meets) of all non-empty finite subsets of any lattice. With additional assumptions, further conclusions may be possible; see Completeness (order theory) for more discussion of this subject. That article also discusses how one may rephrase the above definition in terms of the existence of suitable Galois connections between related posets — an approach of special interest for the category theoretic approach to lattices.

Lattices as algebraic structures

Let L be a set with two binary operations, [\vee] and [\wedge]. A lattice is an algebraic structure [\langle L,\vee,\wedge\rangle] of type [\langle2,2\rangle], such that the following axiomatic identities hold for all members a, b, and c of L:

Commutative laws:		[ a \vee b = b \vee a ]		[ a \wedge b = b \wedge a ]
Associative laws:		[ a \vee (b \vee c) = (a \vee b) \vee c ]		[ a \wedge (b \wedge c) = (a \wedge b) \wedge c ]
Absorption laws:		[ a \vee (a \wedge b) = a ]		[ a \wedge (a \vee b) = a ]

The following important identity follows from the above:

Idempotent laws:

[ a \vee a = a ]

[ a \wedge a = a ]

These axioms assert that (L,[\vee]) and (L,[\wedge]) are each semilattices. The absorption laws, the only equations in which both meet and join appear, distinguish a lattice from a pair of semilattices and assure that the two semilattices interact appropriately. In particular, each semilattice is the dual of the other. A bounded lattice requires that meet and join each have a neutral element, called 1 and 0 by convention. See the entry semilattice.

Lattices have some connections to the groupoid family. Because meet and join both commute and associate, a lattice can be viewed as consisting of two commutative semigroups having the same carrier. If a lattice is bounded, these semigroups are also commutative monoids. The absorption law is the only defining identity that is peculiar to lattice theory.

The closure of L under both meet and join implies, by induction, the existence of the meet and join of any finite subset of L, with one exception: the meet and join of the empty set are the greatest and least elements, respectively. Therefore a lattice contains all finite (including empty) meets and joins only if it is bounded. For this reason, some authors define a lattice so as to require that 0 and 1 be members of L. While definining a lattice in this manner entails no loss of generality, because any lattice can be embedded in a bounded lattice, this definition will not be adopted here.

The algebraic interpretation of lattices plays an essential role in universal algebra.

Connection between the two definitions

The algebraic definition of a lattice implies the order theoretic one, and vice versa.

Obviously, an order-theoretic lattice gives rise to two binary operations [\vee] and [\wedge]. It is easy to see that these operations make (L, [\vee], [\wedge]) into a lattice in the algebraic sense. The converse is true also: Consider an algebraically defined lattice (M, [\vee], [\wedge]). Now define a partial order ≤ on M by setting

x ≤ y if and only if x = x[\wedge]y

or, equivalently,

x ≤ y if and only if y = x[\vee]y

for elements x and y in M. The laws of absorption ensure that both definitions are indeed equivalent. One can now check that the relation ≤ introduced in this way defines a partial ordering within which binary meets and joins are given through the original operations [\vee] and [\wedge]. Conversely, the order induced by the algebraically defined lattice (L, [\vee], [\wedge]) that was derived from the order theoretic formulation above coincides with the original ordering of L.

Since the two definitions of a lattice are equivalent, one may freely invoke aspects of either definition in any way that suits the purpose at hand.

Examples

• For any set A, the collection of all subsets of A (called the power set of A) can be ordered via subset inclusion to obtain a lattice bounded by A itself and the null set. Set intersection and union interpret meet and join, respectively.
• For any set A, the collection of all finite subsets of A, ordered by inclusion, is also a lattice, and will be bounded if and only if A is finite.
• The natural numbers in their usual order form a lattice, under the operations of "min" and "max". 0 is bottom; there is no top.
• The Cartesian square of the natural numbers, ordered by ≤ so that (a,b) ≤ (c,d) ↔ (a ≤ c) & (b ≤ d). (0,0) is bottom; there is no top.
• The non-zero natural numbers also form a lattice under the operations of greatest common divisor and least common multiple operations, with divisibility interpreting the usual order relation. Bottom is 1; there is no top.
• Any complete lattice (also see below) is a (rather specific) bounded lattice. This class gives rise to a broad range of practical examples.
• The set of compact elements of an arithmetic complete lattice is a lattice with a least element, where the lattice operations are given by restricting the respective operations of the arithmetic lattice. This is the specific property which distinguishes arithmetic lattices from algebraic lattices, for which the compacts do only form a join-semilattice. Both of these classes of complete lattices are studied in domain theory.

Further examples are given for each of the additional properties discussed below.

Morphisms of lattices

The appropriate notion of a morphism between two lattices flows easily from the above algebraic definition. Given two lattices (L, [\vee], [\wedge]) and (M, [\cup], [\cap]), a homomorphism of lattices is a function f : L → M such that

f(x[\vee]y) = f(x) [\cup] f(y), and

f(x[\wedge]y) = f(x) [\cap] f(y).

Thus f is a homomorphism of the two underlying semilattices. If the lattices are bounded, then f should preserve the bounds so that:

f(0) = 0, and

f(1) = 1.

In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function preserving binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.

Any homomorphism of lattices is necessarily monotone with respect to the associated ordering relation; see preservation of limits. The converse is of course not true: monotonicity by no means implies the required preservation properties.

Given the standard definition of isomorphisms as invertible morphisms, a lattice isomorphism is just a bijective lattice homomorphism. Lattices and their homomorphisms form a category.

Properties of lattices

We now introduce a number of important properties that lead to interesting special classes of lattices. One, boundedness, has already been discussed.

Completeness

A highly relevant class of lattices are the complete lattices. A lattice is complete if all of its subsets have both a join and a meet, which should be contrasted to the above definition of a lattice where one only requires the existence of all (non-empty) finite joins and meets. Details can be found within the respective article.

Distributivity

Since any lattice comes with two binary operations, it is natural to consider whether one distributes over the other. A lattice (L, [\vee], [\wedge]) is distributive, if the following condition is satisfied for every three elements x, y and z of L:

[x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z)]

This condition is equivalent to the dual statement:

[x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)]

Other characterizations exist, and can be found in the article on distributive lattices. For complete lattices one can formulate various stronger properties, giving rise to the classes of frames and completely distributive lattices. For an overview of these different notions, see distributivity in order theory.

Modularity

Distributivity is too strong a condition for certain applications. A strictly weaker property is modularity: a lattice (L, [\vee], [\wedge]) is modular if, for all elements x, y, and z of L, we have

[x \vee (y \wedge (x \vee z)) = (x \vee y) \wedge (x \vee z)]

Another equivalent statement of this condition is as follows: if x ≤ z then for all y one has

[x \vee (y \wedge z) = (x \vee y) \wedge z]

For example, the lattice of submodules of a module, and the lattice of normal subgroups of a group, all have this special property. Moreover, every distributive lattice is modular.

Continuity and algebraicity

In domain theory, it is natural to seek approximating the elements in a partial order by "much simpler" elements. This leads to the class of continuous posets, consisting of posets where any element can be obtained as the supremum of a directed set of elements that are way-below the element. If one can additionally restrict these to the compact elements of a poset for obtaining these directed sets, then the poset is even algebraic. Both concepts can be applied to lattices as follows:

• A continuous lattice is a complete lattice that is continuous as a poset.
• An algebraic lattice is a complete lattice that is algebraic as a poset.

Both of these classes have interesting properties. For example, continuous lattices can be characterized as algebraic structures (with infinitary operations) satisfying certain identities. While such a characterization is not known for algebraic lattices, they can be described "syntactically" via Scott information systems.

Complements and pseudo-complements

Let L be a bounded lattice with greatest element 1 and least element 0. Two elements x and y of L are complements of each other if and only if:

[x \vee y = 1] and [x \wedge y = 0]

In this case, we write ¬x = y and equivalently, ¬y = x. A bounded lattice for which every element has a complement is called a complemented lattice. The corresponding unary operation over L, called complementation, introduces an analogue of logical negation into lattice theory. The complement is not necessarily unique, nor does it have a special status among all possible unary operations over L. A complemented lattice that is also distributive is a Boolean algebra. For a Boolean algebra, the complement of x is provably unique.

Heyting algebras are an example of distributive lattices having at least some members lacking complements. Every element x of a Heyting algebra has, on the other hand, a pseudo-complement, also denoted ¬x. The pseudo-complement is the greatest element y such that x[\wedge]y = 0. If the pseudo-complement of every element of a Heyting algebra is in fact a complement, then the Heyting algebra is in fact a Boolean algebra.

Sublattices

If L is a lattice and M[\not=\emptyset] is a subset of L such that for every pair of elements a, b in M both a[\wedge]b and a[\vee]b are in M, then we say that M is a sublattice of L.Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. [A Course in Universal Algebra.] Springer-Verlag. ISBN 3540905782.

Free lattices

Using the standard definition of universal algebra, a free lattice over a generating set S is a lattice L together with a function i:S→L, such that any function f from S to the underlying set of some lattice M can be factored uniquely through a lattice homomorphism f° from L to M. Stated differently, for every element s of S we find that f(s) = f°(i(s)) and that f° is the only lattice homomorphism with this property. These conditions basically amount to saying that there is a functor from the category of sets and functions to the category of lattices and lattice homomorphisms which is left adjoint to the forgetful functor from lattices to their underlying sets.

We treat the case of bounded lattices, i.e. algebraic structures with the two binary operations [\vee] and [\wedge] and the two constants (nullary operations) 0 and 1. The set of all correct (well-formed) expressions that can be formulated using these operations on elements from a given set of generators S will be called W(S). This set of words contains many expressions that turn out to be equal in any lattice. For example, if a is some element of S, then a[\vee]1 = 1 and a[\wedge]1 =a. The word problem for lattices is the question, which of these elements have to be identified.

The answer to this problem is as follows. Define a relation <~ on W(S) by setting w <~ v if and only if one of the following holds:

• w = v (this can be restricted to the case where w and v are elements of S),
• w = 0 or v = 1,
• w = w1 [\vee] w2 and both w1<~v and w2<~v hold,
• w = w1 [\wedge] w2 and either w1<~v or w2<~v holds,
• v = v1 [\vee] v2 and either w<~v1 or w<~v2 holds,
• v = v1 [\wedge] v2 and both w<~v1 and w<~v2 hold.

This defines a preorder <~ on W(S). The partially ordered set induced by this preorder (i.e. the set obtained by identifying all words w and v with w<~v and v<~w) is the free lattice on S. The required embedding i is the obvious mapping from a generator a to (the set of words equivalent to) the word a.

One of the consequences of this statement is that the free lattice of a three element set of generators is already infinite. In fact, one can even show that every free lattice on three generators contains a sublattice which is free for a set of four generators. By induction this eventually yields a sublattice free on countably many generators.

The case of lattices that are not bounded is treated similarly, using only the two binary operations in the above construction.

Important lattice-theoretic notions

In the following, let L be a lattice. We define some order-theoretic notions that are of particular importance in lattice theory.

An element x of L is called join-irreducible if and only if

• x = a v b implies x = a or x = b for any a, b in L,
• if L has a 0, x is sometimes required to be different from 0.

When the first condition is generalized to arbitrary joins Va_i, x is called completely join-irreducible. The dual notion is called meet-irreducibility. Sometimes one also uses the terms v-irreducible and ^-irreducible, respectively.

An element x of L is called join-prime if and only if

• x ≤ a v b implies x ≤ a or x ≤ b,
• if L has a 0, x is sometimes required to be different from 0.

Again, this can be generalized to obtain the notion completely join-prime and dualized to yield meet-prime. Any join-prime element is also join-irreducible, and any meet-prime element is also meet-irreducible. If the lattice is distributive the converse is also true.

Other important notions in lattice theory are ideal and its dual notion filter. Both terms describe special subsets of a lattice (or of any partially ordered set in general). Details can be found in the respective articles.

See also

• order theory
• complete lattice
• distributive lattice
• semilattice
• complemented lattice
• orthocomplemented lattice

References

Monographs available free online:

• Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. [A Course in Universal Algebra.] Springer-Verlag. ISBN 3540905782.

• Jipson, P., and H. Rose, [Varieties of Lattices], Lecture Notes in Mathematics 1533, Springer Verlag, 1992. ISBN 0387563148.

Elementary texts recommended for those with limited mathematical maturity:

• Donnellan, Thomas, 1968. Lattice Theory. Pergamon.
• Grätzer, G., 1971. Lattice Theory: First concepts and distributive lattices. W. H. Freeman.

The standard contemporary introductory text:

• Davey, B.A., and H. A. Priestley, 2002. Introduction to Lattices and Order. Cambridge University Press.

The classic advanced monograph:

• Garrett Birkhoff, 1967. Lattice Theory, 3rd ed. Vol. 25 of American Mathematical Society Colloquium Publications. American Mathematical Society.

Free lattices are discussed in the following title, not primarily devoted to lattice theory:

• Johnstone, P.T., 1982. Stone spaces. Cambridge Studies in Advanced Mathematics 3. Cambridge University Press.

Notes

External links

• Eric W. Weisstein et al. "[Lattice.]" From MathWorld--A Wolfram Web Resource.

 

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